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Question

Please explain to me step-by-step how you got the answer. Thank you. 

Find the zeros for the polynomial function and give the multiplicity for each zero. State whether the gragh crosses the x-axis or touches the x-axis and turns around, at each zero. 
13. f(x) = -2(x+2)(x+4)^3 
Answer: -2, multiplicity 1, crosses x-axis; -4, multiplicity 3, crosses x-axis 

Solve the exponential equation. Express the solution set in terms of natural logarithms. 
24. 4^(x+4) = 5^(2x+5) 
Answer: (5 ln 5 - 4 ln 4)/(ln 4 - 2 ln 5) 

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. 
26. log 4 (x-3) + log 4 (x-9) = 2 
Answer: 11 

Write the partical fraction decomposition of the rational expression. 
30. (x-7)/(x-2)(x-3) 
Answer: (5/(x-2)) + (-4/(x-3)) 

Find the sum of the infinite geometric series, if it exists. 
47. (1/3), -1, +3, - . . . 
Answer: Does not exist.

Update : Also, how do you do this problem: 

Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of the minimum or maximum point. 
10. f(x) = -4x^2 - 8x

 

 

Answer 1

• 10).The quadratic function is f ( x ) = - 4x² - 8x.

1. Finding minimum and maximum values of quadratic functions :

   Consider the function with vertex .

   1. If , f  has a minimum at .The minimum value is .

   2. If , f  has a maximum at .The maximum value is .

   The quadratic function is f ( x ) = - 4x ² - 8x.

   Compare the above function with .

   a = - 4, b = - 8, and c = 0.

   a < 0 so, f  has a maximum at .The minimum value is .

   f ( 8/2(-4)) = f ( - 1 ) = - 4(- 1)² - 8(- 1) = - 4 + 8 = 4.

   Then, the maximum value is 4.

   This is a parabola that opens down wards with a maximum of 4 at x = - 1.

   The coordinates of the maximium point is (-1, 4).

Answer 2

Contd......

• 47).

The infinite geometric series is (1/3) - 1 + 3 - ..... .........

First term ( a ) =  1/3

Common ratio ( r ) = 2nd term / 1st term

Common ratio ( r ) =  (- 1)/(1/3) = - 3.

The infinite geometric series

Common ratio ( r ) = - 3

| r | = 3 > 1

So, its impossible to find the sum of the infinite geometric series.

Answer 3

Contd......

• 30).

Let

Taking x terms and constants on both sides we get,

Solve eq (1) and (2) for A and B.

Multiply eq (1) by 2 and write the equations in column form, then subtract them, to find the value of A.

The resultant equation is .

Substitute the value A = 5 in either of the equations (1) and (2), to find the value of B.

Equation (1) : A + B = 1

5 + B = 1

⇒ B = 1 - 5 = - 4.

Therefore, .

Answer 4

Contd......

• 26).

The logarithmic equation is .

Apply Product property of logarithm : .

Apply exponential property of logarithm : ,

By factor by grouping.

Factor :

Apply zero product property.

x -11 = 0 and x - 1= 0

⇒x = 11 and x = 1.

They need to be checked back into the original form of the problem.

For x = 11

.
The above statement is true, so x = 11 is a solution of given equation.
 

For x = 1

, is a mess because we can't take the log of a negative.
So, we can not use x = 1 is a solution.

The exact solution is x = 11.

Answer 5

Contd...........

• 24).

The exponential equation is .

Take natural log on each side.

Apply power property of logarithm :  .

.

The solution is .

Answer 6

• 13).

Repeated zeros :

A factor (x - a)^k , k > 1, yields a repeated zero x = a of multiplicity k.

1 If k is odd, the graph crosses the x - axis at x = a.

2. If k is even, the graph touches the x - axis (but does not cross the x - axis) at x = a.

• The polynomial function is f ( x ) = - 2(x + 2)(x +4)³.

To find the real zeros of the function, set f ( x ) equal to zero and solve for x.

- 2(x + 2)(x +4)³ = 0

Apply zero product property.

x + 2 = 0 and (x +4)³ = 0

⇒ x = - 2 and x = - 4.

So, the real zeros are x = - 2 and x = - 4.

• A factor (x - (- 2))¹ , 1 > 1, not yields a repeated zero x = - 2 of multiplicity 1.
• A factor (x - (- 4))³ , 3 > 1, yields a repeated zero x = - 4 of multiplicity 3.
• At x = - 2, multiplicity is 1(odd), the graph crosses the x - axis at x = - 2.
• At x = - 4, multiplicity is 3(odd), the graph crosses the x - axis at x = - 4.

• Because the function is a fourth degree polynomial, the graph of f can have at most n - 1 = 4 - 1 = 3 turning points.(Turning points, also called relative minima or relative maxima, are points at which the graph changes from increasing to decreasing or vice versa).